American Mathematical Society

Transcription

1 Short Math Guide for L A TEX Michael Downes American Mathematical Society Version 1.09 ( ), currently available at 1. Introduction This is a concise summary of recommended features in L A TEX and a couple of extension packages for writing math formulas. Readers needing greater depth of detail are referred to the sources listed in the bibliography, especially [Lamport], [LUG], [AMUG], [LFG], [LGG], and [LC]. A certain amount of familiarity with standard L A TEX terminology is assumed; if your memory needs refreshing on the L A TEX meaning of command, optional argument, environment, package, and so forth, see [Lamport]. The features described here are available to you if you use L A TEX with two extension packages published by the American Mathematical Society: amssymb and amsmath. Thus, the source file for this document begins with \documentclass{article} \usepackage{amssymb,amsmath} The amssymb package might be omissible for documents whose math symbol usage is relatively modest; the easiest way to test this is to leave out the amssymb reference and see if any math symbols in the document produce Undefined control sequence messages. Many noteworthy features found in other packages are not covered here; see Section 10. Regarding math symbols, please note especially that the list given here is not intended to be comprehensive, but to illustrate such symbols as users will normally find already present in their L A TEX system and usable without installing any additional fonts or doing other setup work. If you have a need for a symbol not shown here, you will probably want to consult The Comprehensive L A TEX Symbols List (Pakin): 2. Inline math formulas and displayed equations 2.1. The fundamentals Entering and leaving math mode in L A TEX is normally done with the following commands and environments. inline formulas $... $ \(... \) displayed equations \[...\] unnumbered \begin{equation*}... \end{equation*} \begin{equation}... \end{equation} unnumbered automatically numbered Note. Alternative environments \begin{math}... \end{math}, \begin{displaymath}... \end{displaymath} are seldom needed in practice. Using the plain TEX notation $$... $$ for displayed equations is not recommended. Although it is not expressly forbidden in LATEX, it is not documented anywhere in the LATEX book as being part of the LATEX command set, and it interferes with the proper operation of various features such as the fleqn option. Environments for handling equation groups and multi-line equations are shown in Table 1. 1

3 Short Math Guide for L A TEX, version 1.09 ( ) Automatic numbering and cross-referencing To get an auto-numbered equation, use the equation environment; to assign a label for cross-referencing, use the \label command: \begin{equation}\label{reio}... \end{equation} To get a cross-reference to an auto-numbered equation, use the \eqref command:... using equations \eqref{ax1} and \eqref{bz2}, we can derive... The above example would produce something like using equations (3.2) and (3.5), we can derive In other words, \eqref{ax1} is equivalent to (\ref{ax1}). To give your equation numbers the form m.n (section-number.equation-number), use the \numberwithin command in the preamble of your document: \numberwithin{equation}{section} For more details on custom numbering schemes see [Lamport, 6.3, C.8.4]. The subequations environment provides a convenient way to number equations in a group with a subordinate numbering scheme. For example, supposing that the current equation number is 2.1, write \begin{equation}\label{first} a=b+c \end{equation} some intervening text \begin{subequations}\label{grp} \begin{align} a&=b+c\label{second}\\ d&=e+f+g\label{third}\\ h&=i+j\label{fourth} \end{align} \end{subequations} to get some intervening text a = b + c (2.9) a = b + c d = e + f + g h = i + j (2.10a) (2.10b) (2.10c) By putting a \label command immediately after \begin{subequations} you can get a reference to the parent number; \eqref{grp} from the above example would produce (2.10) while \eqref{second} would produce (2.10a). 3. Math symbols and math fonts 3.1. Classes of math symbols The symbols in a math formula fall into different classes that correspond more or less to the part of speech each symbol would have if the formula were expressed in words. Certain spacing and positioning cues are traditionally used for the different symbol classes to increase the readability of formulas.

4 Short Math Guide for L A TEX, version 1.09 ( ) 4 Class number Mnemonic Description (part of speech) Examples 0 Ord simple/ordinary ( noun ) A 0 Φ 1 Op prefix operator 2 Bin binary operator (conjunction) + 3 Rel relation/comparison (verb) = < 4 Open left/opening delimiter ( [ { 5 Close right/closing delimiter ) ] } 6 Pun postfix/punctuation., ;! Note 1. The distinction in TEX between class 0 and an additional class 7 has to do only with font selection issues and is immaterial here. Note 2. Symbols of class Bin, notably the minus sign, are automatically coerced to class 0 (no space) if they do not have a suitable left operand. The spacing for a few symbols follows tradition instead of the general rule: although / is (semantically speaking) of class 2, we write k/2 with no space around the slash rather than k / 2. And compare p q p q (no space) with p\mid q p q (class-3 spacing). The proper way to define a new math symbol is discussed in L A TEX 2ε font selection [LFG]. It is not really possible to give a useful synopsis here because one needs first to understand the ramifications of font specifications Some symbols intentionally omitted here The following math symbols that are mentioned in the L A TEX book [Lamport] are intentionally omitted from this discussion because they are superseded by equivalent symbols when the amssymb package is loaded. If you are using the amssymb package anyway, the only thing that you are likely to gain by using the alternate name is an unnecessary increase in the number of fonts used by your document. \Box, see \square \Diamond, see \lozenge \leadsto, see \rightsquigarrow \Join, see \bowtie \lhd, see \vartriangleleft \unlhd, see \trianglelefteq \rhd, see \vartriangleright \unrhd, see \trianglerighteq Furthermore, there are many, many additional symbols available for L A TEX use above and beyond the ones included here. This list is not intended to be comprehensive. For a much more comprehensive list of symbols, including nonmathematically oriented ones such as phonetic alphabetic or dingbats, see The Comprehensive L A TEX Symbols List (Pakin): Latin letters and Arabic numerals The Latin letters are simple symbols, class 0. The default font for them in math formulas is italic. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z a b c d e f g h i j k l m n o p q r s t u v w x y z When adding an accent to an i or j in math, dotless variants can be obtained with \imath and \jmath: ı \imath j \jmath ĵ \hat{\jmath} Arabic numerals 0 9 are also of class 0. Their default font is upright/roman

9 Short Math Guide for L A TEX, version 1.09 ( ) 9 \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\esssup}{ess\,sup} one could write \rank(x) rank(x) \esssup(y,z) ess sup(y, z) The star form \DeclareMathOperator* creates an operator that takes limits in a displayed formula like sup or max. When predefining such a named operator is problematic (e.g., when using one in the title or abstract of an article), there is an alternative form that can be used directly: \operatorname{rank}(x) rank(x) Math font switches Not all of the fonts necessary to support comprehensive math font switching are commonly available in a typical L A TEX setup. Here are the results of applying various font switches to a wide range of math symbols when the standard set of Computer Modern fonts is in use. It can be seen that the only symbols that respond correctly to all of the font switches are the uppercase Latin letters. In fact, nearly all math symbols apart from Latin letters remain unaffected by font switches; and although the lowercase Latin letters, capital Greek letters, and numerals do respond properly to some font switches, they produce bizarre results for other font switches. (Use of alternative math font sets such as Lucida New Math may ameliorate the situation somewhat.) default \mathbf \mathsf \mathit \mathcal \mathbb \mathfrak X X X X X X X x x x x x [ ] [ ] [ ] [ ] [ ] [ ] [ ] = = = = = = = Ξ Ξ Ξ Ξ g ξ ξ ξ ξ ξ ξ ξ ℵ ℵ ℵ ℵ ℵ ℵ ℵ R R R R R R R A common desire is to get a bold version of a particular math symbol. For those symbols where \mathbf is not applicable, the \boldsymbol or \pmb commands can be used. A + πa 0 A + πa 0 A + πa 0 (3.1) A_\infty + \pi A_0 \sim \mathbf{a}_{\boldsymbol{\infty}} \boldsymbol{+} \boldsymbol{\pi} \mathbf{a}_{\boldsymbol{0}} \sim\pmb{a}_{\pmb{\infty}} \pmb{+}\pmb{\pi} \pmb{a}_{\pmb{0}} The \boldsymbol command is obtained preferably by using the bm package, which provides a newer, more powerful version than the one provided by the amsmath package. Generally speaking, it is ill-advised to apply \boldsymbol to more than one symbol at a time.

11 Short Math Guide for L A TEX, version 1.09 ( ) 11 \bigl( \begin{smallmatrix} a&b\\ c&d \end{smallmatrix} \bigr) To produce a row of dots in a matrix spanning a given number of columns, use \hdotsfor. For example, \hdotsfor{3} in the second column of a four-column matrix will print a row of dots across the final three columns. For piece-wise function definitions there is a cases environment: P_{r-j}=\begin{cases} 0& \text{if $r-j$ is odd},\\ r!\,(-1)^{(r-j)/2}& \text{if $r-j$ is even}. \end{cases} Notice the use of \text and the embedded math. Note. The plain TEX form \matrix{...\cr...\cr} and the related commands \pmatrix, \cases should be avoided in LATEX (and when the amsmath package is loaded they are disabled) Math spacing commands When the amsmath package is used, all of these math spacing commands can be used both in and out of math mode. Abbrev. Spelled out Example Abbrev. Spelled out Example no space 34 no space 34 \, \thinspace 3 4 \! \negthinspace 34 \: \medspace 3 4 \negmedspace 34 \; \thickspace 3 4 \negthickspace 34 \quad 3 4 \qquad 3 4 For finer control over math spacing, use \mspace and math units. One math unit, or mu, is equal to 1/18 em. Thus to get a negative half \quad write \mspace{-9.0mu}. There are also three commands that leave a space equal to the height and/or width of a given fragment of L A TEX material: Example Result \phantom{xxx} space as wide and high as three X s \hphantom{xxx} space as wide as three X s; height 0 \vphantom{x} space of width 0, height = height of X 4.6. Dots For preferred placement of ellipsis dots (raised or on-line) in various contexts there is no general consensus. It may therefore be considered a matter of taste. By using the semantically oriented commands \dotsc for dots with commas \dotsb for dots with binary operators/relations \dotsm for multiplication dots \dotsi for dots with integrals \dotso for other dots (none of the above) instead of \ldots and \cdots, you make it possible for your document to be adapted to different conventions on the fly, in case (for example) you have to submit it to a publisher who insists on following house tradition in this respect. The default treatment for the various kinds follows American Mathematical Society conventions: We have the series $A_1,A_2,\dotsc$, the regional sum $A_1+A_2+\dotsb$, the orthogonal product $A_1A_2\dotsm$, and the infinite integral \[\int_{a_1}\int_{a_2}\dotsi\]. We have the series A 1, A 2,..., the regional sum A 1 + A 2 +, the orthogonal product A 1 A 2, and the infinite integral. A 2 A 1

12 Short Math Guide for L A TEX, version 1.09 ( ) Nonbreaking dashes The command \nobreakdash suppresses the possibility of a linebreak after the following hyphen or dash. For example, if you write pages 1 9 as pages 1\nobreakdash--9 then a linebreak will never occur between the dash and the 9. You can also use \nobreakdash to prevent undesirable hyphenations in combinations like $p$-adic. For frequent use, it s advisable to make abbreviations, e.g., \newcommand{\p}{$p$\nobreakdash}% for "\p-adic" \newcommand{\ndash}{\nobreakdash\textendash}% for "pages 1\Ndash 9" % For "\n dimensional" ("n-dimensional"): \newcommand{\n}[1]{$n$\nobreakdash-\hspace{0pt}} The last example shows how to prohibit a linebreak after the hyphen but allow normal hyphenation in the following word. (It suffices to add a zero-width space after the hyphen.) 4.8. Roots The command \sqrt produces a square root. To specify an alternate radix give an optional argument. n \sqrt{\frac{n}{n-1} S} n 1 S, \sqrt[3]{2} Boxed formulas The command \boxed puts a box around its argument, like \fbox except that the contents are in math mode: η C(δ(η) + Λ M (0, δ)) (4.2) \boxed{\eta \leq C(\delta(\eta) +\Lambda_M(0,\delta))} If you need to box an equation including the equation number, see the FAQ that comes with the amsmath package. 5. Fractions and related constructions 5.1. The \frac, \dfrac, and \tfrac commands The \frac command takes two arguments numerator and denominator and typesets them in normal fraction form. Use \dfrac or \tfrac to overrule L A TEX s guess about the proper size to use for the fraction s contents (t = text-style, d = display-style). \begin{equation} \frac{1}{k}\log_2 c(f)\;\tfrac{1}{k}\log_2 c(f)\; \end{equation} 1 k log 2 c(f) 1 k log 2 c(f) (5.1) nπ θ + ψ Rz = 2 ( ) 2 ( ) θ + ψ log B 2. (5.2) A \begin{equation} \Re{z} =\frac{n\pi \dfrac{\theta +\psi}{2}}{ \left(\dfrac{\theta +\psi}{2}\right)^2 + \left( \dfrac{1}{2} \log \left\lvert\dfrac{b}{a}\right\rvert\right)^2}. \end{equation} 5.2. ( The \binom, \dbinom, and \tbinom commands For binomial expressions such as n ) k there are \binom, \dbinom and \tbinom commands: ( ) ( ) k k 2 k 2 k k 2 (5.3) 1 2 2^k-\binom{k}{1}2^{k-1}+\binom{k}{2}2^{k-2}

13 Short Math Guide for L A TEX, version 1.09 ( ) The \genfrac command The capabilities of \frac, \binom, and their variants are subsumed by a generalized fraction command \genfrac with six arguments. The last two correspond to \frac s numerator and denominator; the first two are optional delimiters (as seen in \binom); the third is a line thickness override (\binom uses this to set the fraction line thickness to 0 pt i.e., invisible); and the fourth argument is a mathstyle override: integer values 0 3 select respectively \displaystyle, \textstyle, \scriptstyle, and \scriptscriptstyle. If the third argument is left empty, the line thickness defaults to normal. \genfrac{left-delim}{right-delim}{thickness} {mathstyle}{numerator}{denominator} To illustrate, here is how \frac, \tfrac, and \binom might be defined. \newcommand{\frac}[2]{\genfrac{}{}{}{}{#1}{#2}} \newcommand{\tfrac}[2]{\genfrac{}{}{}{1}{#1}{#2}} \newcommand{\binom}[2]{\genfrac{(}{)}{0pt}{}{#1}{#2}} Note. For technical reasons, using the primitive fraction commands \over, \atop, \above in a LATEX document is not recommended (see, e.g., amsmath.faq) Continued fractions The continued fraction (5.4) can be obtained by typing \cfrac{1}{\sqrt{2}+ \cfrac{1}{\sqrt{2}+ \cfrac{1}{\sqrt{2}+\dotsb }}} This produces better-looking results than straightforward use of \frac. Left or right placement of any of the numerators is accomplished by using \cfrac[l] or \cfrac[r] instead of \cfrac. 6. Delimiters 6.1. Delimiter sizes Unless you indicate otherwise, delimiters in math formulas will remain at the standard size regardless of the height of the enclosed material. To get larger sizes, you can either select a particular size using a \big... prefix (see below), or you can use \left and \right prefixes for autosizing. The automatic delimiter sizing done by \left and \right has two limitations: First, it is applied mechanically to produce delimiters large enough to encompass the largest contained item, and second, the range of sizes has fairly large quantum jumps. This means that an expression that is infinitesimally too large for a given delimiter size will get the next larger size, a jump of 6pt or so (3pt top and bottom) in normal-sized text. There are two or three situations where the delimiter size is commonly adjusted. These adjustments are done using the following commands: Delimiter text \left \bigl \Bigl \biggl \Biggl size size \right \bigr \Bigr \biggr \Biggr Result (b)( c ( c ) d ) (b) ( )( c ) ( )( ( )( ) c c b b b d d d) ) c b)( d d

14 Short Math Guide for L A TEX, version 1.09 ( ) 14 The first kind of adjustment is done for cumulative operators with limits, such as summation signs. With \left and \right the delimiters usually turn out larger than necessary, and using the Big or bigg sizes instead gives better results: i a i j x ij p 1/p versus [ ] p 1/p a i x ij \biggl[\sum_i a_i\bigl\lvert\sum_j x_{ij}\bigr\rvert^p\biggr]^{1/p} The second kind of situation is clustered pairs of delimiters where \left and \right make them all the same size (because that is adequate to cover the encompassed material) but what you really want is to make some of the delimiters slightly larger to make the nesting easier to see. ( ((a 1 b 1 ) (a 2 b 2 )) ((a 2 b 1 ) + (a 1 b 2 )) versus (a1 b 1 ) (a 2 b 2 ) )( (a 2 b 1 ) + (a 1 b 2 ) ) \left((a_1 b_1) - (a_2 b_2)\right) \left((a_2 b_1) + (a_1 b_2)\right) \quad\text{versus}\quad \bigl((a_1 b_1) - (a_2 b_2)\bigr) \bigl((a_2 b_1) + (a_1 b_2)\bigr) The third kind of situation is a slightly oversize object in running text, such as b d where the delimiters produced by \left and \right cause too much line spreading. In that case \bigl and \bigr can be used to produce delimiters that are larger than the base size but still able to fit within the normal line spacing: b. d 6.2. Vertical bar notations The use of the character to produce paired delimiters is not recommended. There is an ambiguity about the directionality of the symbol that will in rare cases produce incorrect spacing e.g., k = -k produces k = k. Using \lvert for a left vert bar and \rvert for a right vert bar whenever they are used in pairs will prevent this problem: compare k, produced by \lvert -k\rvert. For double bars there are analogous \lvert, \rvert commands. Recommended practice is to define suitable commands in the document preamble for any paired-delimiter use of vert bar symbols: \providecommand{\abs}[1]{\lvert#1\rvert} \providecommand{\norm}[1]{\lvert#1\rvert} whereupon \abs{z} would produce z and \norm{v} would produce v. 7. The \text command The main use of the command \text is for words or phrases in a display. It is similar to \mbox in its effects but, unlike \mbox, automatically produces subscript-size text if used in a subscript. f_{[x_{i-1},x_i]} \text{ is monotonic,} \quad i = 1,\dots,c+1 i f [xi 1,x i] is monotonic, i = 1,..., c + 1 (7.1) 7.1. \mod and its relatives Commands \mod, \bmod, \pmod, \pod deal with the special spacing conventions of mod notation. \mod and \pod are variants of \pmod preferred by some authors; \mod omits the parentheses, whereas \pod omits the mod and retains the parentheses. gcd(n, m mod n); x y (mod b); x y mod c; x y (d) (7.2) \gcd(n,m\bmod n);\quad x\equiv y\pmod b ;\quad x\equiv y\mod c;\quad x\equiv y\pod d j

15 Short Math Guide for L A TEX, version 1.09 ( ) Integrals and sums 8.1. Altering the placement of limits The limits on integrals, sums, and similar symbols are placed either to the side of or above and below the base symbol, depending on convention and context. L A TEX has rules for automatically choosing one or the other, and most of the time the results are satisfactory. In the event they are not, there are three L A TEX commands that can be used to influence the placement of the limits: \limits, \nolimits, \displaylimits. Compare z 6 (t)φ(x) z 6 (t)φ(x) x x z(t) <X 0 and \int_{\abs{x-x_z(t)}<x_0}... x x z(t) <X 0 \int\limits_{\abs{x-x_z(t)}<x_0}... The \limits command should follow immediately after the base symbol to which it applies, and its meaning is: shift the following subscript and/or superscript to the limits position, regardless of the usual convention for this symbol. \nolimits means to shift them to the side instead, and \displaylimits, which might be used in defining a new kind of base symbol, means to use standard positioning as for the \sum command. See also the description of the intlimits and nosumlimits options in [AMUG] Multiple integral signs \iint, \iiint, and \iiiint give multiple integral signs with the spacing between them nicely adjusted, in both text and display style. \idotsint is an extension of the same idea that gives two integral signs with dots between them. f(x, y) dx dy f(x, y, z) dx dy dz (8.1) A A A f(w, x, y, z) dw dx dy dz A f(x 1,..., x k ) (8.2) 8.3. Multiline subscripts and superscripts The \substack command can be used to produce a multiline subscript or superscript: for example \sum_{\substack{ P (i, j) 0\le i\le m\\ 0 i m 0<j<n 0<j<n}} P(i,j) 8.4. The \sideset command There s also a command called \sideset, for a rather special purpose: putting symbols at the subscript and superscript corners of a symbol like or. Note: The \sideset command is only designed for use with large operator symbols; with ordinary symbols the results are unreliable. With \sideset, you can write \sideset{}{ } \sum_{n<k,\;\text{$n$ odd}} ne_n n<k, n odd The extra pair of empty braces is explained by the fact that \sideset has the capability of putting an extra symbol or symbols at each corner of a large operator; to put an asterisk at each corner of a product symbol, you would type \sideset{_*^*}{_*^*}\prod ne n

16 Short Math Guide for L A TEX, version 1.09 ( ) Changing the size of elements in a formula The L A TEX mechanisms for changing font size inside a math formula are completely different from the ones used outside math formulas. If you try to make something larger in a formula with one of the text commands such as \large or \huge: # {\large \#} you will get a warning message Command \large invalid in math mode Such an attempt, however, often indicates a misunderstanding of how L A TEX math symbols work. If you want a # symbol analogous to a summation sign in its typographical properties, then in principle the best way to achieve that is to define it as a symbol of type mathop with the standard L A TEX \DeclareMathSymbol command (see [LFG]). [In this particular example it is currently unlikely that you will be able to lay your hands on a math font with a suitable text-size/display-size pair, but that is probably best understood as a problem of inadequate fonts, not as a L A TEX problem.] Consider the expression: n>0 zn 1 k n (1 qk ) \frac{\sum_{n > 0} z^n} {\prod_{1\leq k\leq n} (1-q^k)} Using \dfrac instead of \frac wouldn t change anything in this case; if you want the sum and product symbols to appear full size, you need the \displaystyle command: n>0 1 k n z n (1 q k ) \frac{{\displaystyle\sum_{n > 0} z^n}} {{\displaystyle\prod_{1\leq k\leq n} (1-q^k)}} And if you want full-size symbols but with limits on the side, use the nolimits command also: n>0 zn \frac{{\displaystyle\sum\nolimits_{n> 0} z^n}} (1 1 k n qk ) {{\displaystyle\prod\nolimits_{1\leq k\leq n} (1-q^k)}} There are similar commands \textstyle, \scriptstyle, and \scriptscriptstyle, to force L A TEX to use the symbol size and spacing that would be applied in (respectively) inline math, first-order subscript, or second-order order subscript, even when the current context would normally yield some other size. Note: These commands belong to a special class of commands referred to in the L A TEX book as declarations. In particular, notice where the braces fall that delimit the effect of the command: Right:{\displaystyle...}Wrong:\displaystyle{...} 10. Other packages of interest Many other L A TEX packages that address some aspect of mathematical formulas are available from CTAN (the Comprehensive TEX Archive Network). To recommend a few examples: accents Under accents and accents using arbitrary symbols. amsthm General theorem and proof setup. bm Bold math package, provides a more general and more robust implementation of \boldsymbol. cases Apply a large brace to two or more equations without losing the individual equation numbers.

Short Math Guide for L A TEX Michael Downes American Mathematical Society Version 1.07 (2000/07/19) 1. Introduction This is a concise summary of recommended features in L A TEX and a couple of extension

Using Word 2007 s new equation editor Open a new equation by typing ALT = You should see something like this: You can now insert symbols using the Equation toolbar that appears. Word now also implements

L A TEX and AMS-L A TEX Symbols March 17, 2008 I About This Document LATEX This document lists symbols in standard LaTeX, AMS-L A TEX and a few additional packages. The document is optimized for viewing

Vectors & Motion Representing Motion Four Types of Motion We ll Study What is a vector? What is a vector? A quantity with magnitude & direction Quiz 1. What is the difference between speed and velocity?

22 MAPS 34 Aditya Mahajan Display Math in ConTEXt ConTEXt rehab for amsmath addicts Abstract This article explains how to do various kinds of alignments in ConTEXt. A visual output is presented, and it

A Beginner s Guide to L A TEX David Xiao dxiao@cs.princeton.edu September 12, 2005 1 Introduction L A TEX is the standard mathematical typesetting program. This document is for people who have never used

Tips for Using MyMathLab Plus Navigating the Text 1. To open you course section plan, select the MTE you are enrolled in. Then click on the section you wish to complete. Click on the section you wish to

Math into L A TEX An Introduction to L A TEX and AMS-L A TEX This book is dedicated to those who worked so hard and for so long to bring these important tools to us: The L A TEX3 team and in particular

Microsoft Word 2016 Tutorial This tutorial requires a basic understanding of how to use Microsoft Word and focuses only on operations useful for the Word and Excel Assignment 2016. This transcript is supplementary

Getting Started Guide Chapter 9 Getting Started with Math OpenOffice.org's Equation Editor Copyright This document is Copyright 2005 2008 by its contributors as listed in the section titled Authors. You

MS-Word File with Mathematical Symbols First I give a list of symbols for both MS-Word and Powerpoint. Then I explain how to get summation and integration, how to put one thing above another, and, finally,

OpenOffice.org HowTo: Adapted and translated from OpenOffice.org Formel This version:1.0 24 July 2005 First Edition: Unknown First English Edition: 05 June 2003 Contents Contents Table of Contents Contents...2

Mathematical and scientific symbols September 2014 There are several techniques for writing maths and science expressions when using Digital Question papers: 1. hand-write on the paper (either on the hard

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

32 Useful Mathematical Symbols Symbol What it is How it is read How it is used Sample expression + * ddition sign OR Multiplication sign ND plus or times and x Multiplication sign times Sum of a few disjunction

l TELECOMMUNICATION ENGINEERING UNIVERSITY OF TWENTE University of Twente Department of Electrical Engineering Chair for Telecommunication Engineering Guide for writing assignment reports by A.B.C. Surname

EQUATION EDITOR A mathematical typesetting program for Macintosh & PC computers. Equation Editor is bundled with Microsoft Word and is therefore protected by licensing agreements and copyright statutes.

The Word 2007/2010 Equation Editor Contents The Word 2007/2010 Equation Editor... 1 When the Equation Editor Should Be Used... 1 Why the Equation Editor Should Be Used... 1 How to Enter the Equation Editor

Writer Guide Chapter 16 Math Objects The OpenOffice.org Equation Editor Copyright This document is Copyright 2008 by its contributors as listed in the section titled Authors. You may distribute it and/or

.6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

Math Guide The OpenOffice.org Equation Editor This PDF is designed to be read onscreen, two pages at a time. If you want to print a copy, your PDF viewer should have an option for printing two pages on

About L A T E X L A T E X Tutorial You can either print this document or follow it on line. L A T E X (pronounced either Lay-tech or Lah-tech ) is a portable document formatting system based on T E X (pronounced

TI-86 Graphing Calculator Keystroke Guide In your textbook you will notice that on some pages a key-shaped icon appears next to a brief description of a feature on your graphing calculator. In this guide

url.sty version 3.4 Donald Arseneau 2013-09-16 The package defines a form of \verb command that allows linebreaks at certain characters or combinations of characters, accepts reconfiguration, and can usually

INTRODUCTION TO EXCEL 1 INTRODUCTION Anyone who has used a computer for more than just playing games will be aware of spreadsheets A spreadsheet is a versatile computer program (package) that enables you

TI-83 Plus Graphing Calculator Keystroke Guide In your textbook you will notice that on some pages a key-shaped icon appears next to a brief description of a feature on your graphing calculator. In this

1 Matlab 1) Fundamentals a) Getting Help for more detailed help on any topic, typing help, then a space, and then the matlab command brings up a detailed page on the command or topic. For really difficult

VISUAL GUIDE to RX Scripting for Roulette Xtreme - System Designer 2.0 UX Software - 2009 TABLE OF CONTENTS INTRODUCTION... ii What is this book about?... iii How to use this book... iii Time to start...

The Title of a TU Wien Report Dr. Samuel Author The Date I. The First Part 1 1 Sample Mathematics and Text 1.1 In-line and Displayed Mathematics The expression P 1 i=1 a i is in-line mathematics, while

SMART Notebook Math Tools User s Guide The content of this guide is provided for informational purposes only and is subject to change without notice. Trademark Notice SMART Board, SMART Notebook, the SMART

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

OBJECTIVE We will practice using formulas and functions in. ESSENTIAL SKILLS Enter formulas by typing Enter formulas by Point mode Apply the AVERAGE, MAX, and MIN functions Verify a formula using Range

BLACKBOARD 9.1: Text Editor The text editor in Blackboard is a feature that appears in many different areas, but generally has the same look and feel no matter where it appears. The text editor has changed

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started

C A S I O f x - 8 2 A U UNIVERSITY OF SOUTHERN QUEENSLAND The Learning Centre Learning and Teaching Support Unit Mastering the calculator using the Casio fx-82au Learning and Teaching Support Unit (LTSU)

Quick Start Instructions on using MyEconLab with the JAWS Screen Reader* To work on assignments in MyEconLab with JAWS, follow these steps to turn on the Automatically Detect for Accessibility Setting:

Table of Contents Elements of An Excel Document... 2 Resizing and Hiding Columns and Rows... 3 Using Panes to Create Spreadsheet Headers... 3 Using the AutoFill Command... 4 Using AutoFill for Sequences...

Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

A THESIS/DISSERTATION FORMATTING MANUAL FOR THE PURDUE UNIVERISTY CHEMISTRY DEPARTMENT A Manual Submitted to the Faculty and Graduate Students of Purdue University by Arwen N. Revis To help fulfill the

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition

C A S I O f x - 8 2 M S MASTERING THE CALCULATOR USING THE CASIO fx-82ms Learning and Teaching Support Unit (LTSU) The Learning Centre Guide book Written by Linda Galligan Published by University of Southern

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

Introduction to mathematical arguments (background handout for courses requiring proofs) by Michael Hutchings A mathematical proof is an argument which convinces other people that something is true. Math

Advanced Higher Mathematics Course Assessment Specification (C747 77) Valid from August 2015 This edition: April 2016, version 2.4 This specification may be reproduced in whole or in part for educational

XMGrace Fancy characters and stuff In XMGrace it is possible to write Greek letters, do superscripts and subscripts and the like. This tex-file/pdf will hopefully keep a list of what I have learnt (starting

Introduction to Mathcad Mathcad is as versatile and powerful as programming languages, yet it is as easy to learn as a spreadsheet! This text will introduce you to: Fundamental functions of Mathcad, including

Ribbon menu The Ribbon menu system with tabs for various Excel commands. This Ribbon system replaces the traditional menus used with Excel 2003. Above the Ribbon in the upper-left corner is the Microsoft

List of Standard Counting Problems Advanced Problem Solving This list is stolen from Daniel Marcus Combinatorics a Problem Oriented Approach (MAA, 998). This is not a mathematical standard list, just a

Generated using version 3.2 of the official AMS L A TEX template 1 A Sample American Meteorological Society L A TEX Document 2 Brian Papa, Jack Crielson, and Sarah Cooley American Meteorological Society,